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Re: [Phys-l] The "why" questions



On 12/3/2010 12:11 PM, John Mallinckrodt wrote:
On Dec 3, 2010, at 10:04 AM, brian whatcott wrote:
The CofMass shows a ramp of acceleration which settles (finally) on f/m.
Not so. Assuming f is the net force on some well-defined object or system of objects of total mass m, then the acceleration of the center of mass is identical to f/m at all times. This is easy to show. It seems to me that Al and Jeffrey have correctly distinguished between that acceleration and the acceleration of a particle that may be at or near the center of mass. That particle's motion depends on speed of sound considerations. Not so for the motion of the center of mass.

John Mallinckrodt
Cal Poly Pomona


Hmmm...pleasant though it is to tweak the physics tail, so to speak, I had better pay more attention. It is the case that when that article of faith, f=m.a is amended so as to apply to a mass considered to act at a representative point, some intransitive difficulties are resolved, so perhaps I should try harder now that I have strayed into that even more holy place called conservation of momentum.

It is an insight, pleasing even to high-schoolers, to find that dealing with colliding masses of various elastic and plastic kinds can be dealt with handily by this rule.

I wonder if it has limits?
Let's suppose I am dealing with a long fairly thin rod extruded in a way that inhibits awkward transverse vibrations, and I apply a sudden onset uniform force to one end. Of course the rod cannot move bodily at the onset, and I expect for one skilled in the art, it is not difficult to sum an expression for an adjacent material layer times its velocity over the length to arrive at an expression that upholds the desired "instantaneous" form of acceleration for that central point.

We would ( I suppose) find agreement that before the proximal end begins to move bodily (rather than just initiating a compression wave along its length) intelligence of the act needs to traverse the length in both directions. So now I suppose that the distal end is held fast behind a massive bulwark.
When the compression wave reaches this "immovable object" it meets either of two fates:
1) The compression wave reflects so that the compression doubles.
2) the plastic limit of the material is so close to the magnitude of the initial compression, that the end yields plastically, and little additional compression returns to the proximal end.

In case 1) especially, the rod is now bodily compressed axially, end to end, and with further compression from the reflected wave, the distal end now shortens, rebounding from the massive barrier.

In either case, either most or some of the initial compression relaxes, so that the force is displaced 'backwards', returning some energy in this way.
It is a familiar observation that a hammer blow usually causes the hammer head to rebound,
and the far end of say a drift or chisel jumps back.
This is the very mechanism which I am describing though perhaps in cartoon form.
One even uses this momentum reflection in three axes to develop an easy embodiment of the statistical form of a thermal expression known to a patent clerk of repute.

Now I need to convince myself of the justice of John M's upholding of the conservation of momentum. After all, it appears to work quite well for pieces of plasticene!

Perhaps he has it that the deltaM at the proximal end of the rod moves swiftly at first, (as some other contributor suggested was the case?) and as the wave travels, the instantaneous velocity decreases, so that when the compression wave reaches the far end, its wave velocity is slowest, while the summation produces a constant m.a for a point at the center of mass.
No, John would not visualize a decaying velocity. Perhaps it is a constant wave velocity down the rod, with a linearly increasing length of the rod being compressed, so that the distal end is accelerating at an increasing rate. No, that cannot be the case.

This evidently contrasts with the electric cell applied to the end of a transmission line. There the amplitude of the wave is constant, and the wavefront velocity is also sensibly constant.

Perhaps it is inappropriate to be using a transmission line as a metaphor for a force applied to an extended mass? I will pause here, because this pretend experimentation is beginning to ramble unneccessarily.

Brian W